Cubic critical, wide range

Percentage Accurate: 18.3% → 99.4%

Time: 13.4s

Alternatives: 7

Speedup: 2.9×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 18.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(a \cdot -3, c, 0\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}}{a \cdot 3} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ (fma (* a -3.0) c 0.0) (+ b (sqrt (fma c (* a -3.0) (* b b)))))
  (* a 3.0)))

C

double code(double a, double b, double c) {
	return (fma((a * -3.0), c, 0.0) / (b + sqrt(fma(c, (a * -3.0), (b * b))))) / (a * 3.0);
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(Float64(a * -3.0), c, 0.0) / Float64(b + sqrt(fma(c, Float64(a * -3.0), Float64(b * b))))) / Float64(a * 3.0))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(a * -3.0), $MachinePrecision] * c + 0.0), $MachinePrecision] / N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in c around inf

   \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{c \cdot \left(\frac{{b}^{2}}{c} - 3 \cdot a\right)}}}{3 \cdot a} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{c \cdot \left(\frac{{b}^{2}}{c} - 3 \cdot a\right)}}}{3 \cdot a} \]

   2. cancel-sign-sub-inv N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{c \cdot \color{blue}{\left(\frac{{b}^{2}}{c} + \left(\mathsf{neg}\left(3\right)\right) \cdot a\right)}}}{3 \cdot a} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{c \cdot \left(\frac{{b}^{2}}{c} + \color{blue}{-3} \cdot a\right)}}{3 \cdot a} \]

   4. +-commutative N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{c \cdot \color{blue}{\left(-3 \cdot a + \frac{{b}^{2}}{c}\right)}}}{3 \cdot a} \]

   5. *-commutative N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{c \cdot \left(\color{blue}{a \cdot -3} + \frac{{b}^{2}}{c}\right)}}{3 \cdot a} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{c \cdot \color{blue}{\mathsf{fma}\left(a, -3, \frac{{b}^{2}}{c}\right)}}}{3 \cdot a} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{c \cdot \mathsf{fma}\left(a, -3, \color{blue}{\frac{{b}^{2}}{c}}\right)}}{3 \cdot a} \]

   8. unpow2 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{c \cdot \mathsf{fma}\left(a, -3, \frac{\color{blue}{b \cdot b}}{c}\right)}}{3 \cdot a} \]

   9. lower-*.f64 17.5

      \[\leadsto \frac{\left(-b\right) + \sqrt{c \cdot \mathsf{fma}\left(a, -3, \frac{\color{blue}{b \cdot b}}{c}\right)}}{3 \cdot a} \]

5. Applied rewrites 17.5%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{c \cdot \mathsf{fma}\left(a, -3, \frac{b \cdot b}{c}\right)}}}{3 \cdot a} \]

7. Applied rewrites 18.0%

   \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(-b\right)}}}{3 \cdot a} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\left(c \cdot \color{blue}{\left(a \cdot -3\right)} + b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\left(c \cdot \left(a \cdot -3\right) + \color{blue}{b \cdot b}\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\left(c \cdot \left(a \cdot -3\right) + b \cdot b\right) - \color{blue}{b \cdot b}}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   4. associate--l+ N/A

      \[\leadsto \frac{\frac{\color{blue}{c \cdot \left(a \cdot -3\right) + \left(b \cdot b - b \cdot b\right)}}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\frac{c \cdot \color{blue}{\left(a \cdot -3\right)} + \left(b \cdot b - b \cdot b\right)}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   6. *-commutative N/A

      \[\leadsto \frac{\frac{c \cdot \color{blue}{\left(-3 \cdot a\right)} + \left(b \cdot b - b \cdot b\right)}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{c \cdot \left(\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a\right) + \left(b \cdot b - b \cdot b\right)}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   8. distribute-lft-neg-in N/A

      \[\leadsto \frac{\frac{c \cdot \color{blue}{\left(\mathsf{neg}\left(3 \cdot a\right)\right)} + \left(b \cdot b - b \cdot b\right)}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{c \cdot \left(\mathsf{neg}\left(\color{blue}{3 \cdot a}\right)\right) + \left(b \cdot b - b \cdot b\right)}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   10. *-commutative N/A

       \[\leadsto \frac{\frac{\color{blue}{\left(\mathsf{neg}\left(3 \cdot a\right)\right) \cdot c} + \left(b \cdot b - b \cdot b\right)}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   11. +-inverses N/A

       \[\leadsto \frac{\frac{\left(\mathsf{neg}\left(3 \cdot a\right)\right) \cdot c + \color{blue}{0}}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   12. lower-fma.f64 N/A

       \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(3 \cdot a\right), c, 0\right)}}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{3 \cdot a}\right), c, 0\right)}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   14. distribute-lft-neg-in N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot a}, c, 0\right)}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   15. metadata-eval N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{-3} \cdot a, c, 0\right)}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   16. *-commutative N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{a \cdot -3}, c, 0\right)}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   17. lift-*.f64 99.5

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{a \cdot -3}, c, 0\right)}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(-b\right)}}{3 \cdot a} \]

9. Applied rewrites 99.5%

   \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(a \cdot -3, c, 0\right)}}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(-b\right)}}{3 \cdot a} \]

10. Final simplification 99.5%

    \[\leadsto \frac{\frac{\mathsf{fma}\left(a \cdot -3, c, 0\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}}{a \cdot 3} \]
11. Add Preprocessing

Alternative 2: 95.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}, a, \frac{c}{b \cdot -2}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma (/ (* (* c c) -0.375) (* b (* b b))) a (/ c (* b -2.0))))

C

double code(double a, double b, double c) {
	return fma((((c * c) * -0.375) / (b * (b * b))), a, (c / (b * -2.0)));
}

Julia

function code(a, b, c)
	return fma(Float64(Float64(Float64(c * c) * -0.375) / Float64(b * Float64(b * b))), a, Float64(c / Float64(b * -2.0)))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(c / N[(b * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto c \cdot \color{blue}{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}}\right) + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)} \]

   3. associate-*r/ N/A

      \[\leadsto c \cdot \color{blue}{\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{3}}} + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto c \cdot \frac{\color{blue}{\left(\frac{-3}{8} \cdot a\right) \cdot c}}{{b}^{3}} + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \]

   5. associate-*l/ N/A

      \[\leadsto c \cdot \color{blue}{\left(\frac{\frac{-3}{8} \cdot a}{{b}^{3}} \cdot c\right)} + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto c \cdot \left(\color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right)} \cdot c\right) + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \]

   7. distribute-lft-in N/A

      \[\leadsto \color{blue}{c \cdot \left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \color{blue}{c \cdot \left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

   9. associate-*r/ N/A

      \[\leadsto c \cdot \left(\color{blue}{\frac{\frac{-3}{8} \cdot a}{{b}^{3}}} \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   10. associate-*l/ N/A

       \[\leadsto c \cdot \left(\color{blue}{\frac{\left(\frac{-3}{8} \cdot a\right) \cdot c}{{b}^{3}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   11. associate-*r* N/A

       \[\leadsto c \cdot \left(\frac{\color{blue}{\frac{-3}{8} \cdot \left(a \cdot c\right)}}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   12. associate-*r/ N/A

       \[\leadsto c \cdot \left(\color{blue}{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto c \cdot \left(\color{blue}{\frac{a \cdot c}{{b}^{3}} \cdot \frac{-3}{8}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   14. associate-/l* N/A

       \[\leadsto c \cdot \left(\color{blue}{\left(a \cdot \frac{c}{{b}^{3}}\right)} \cdot \frac{-3}{8} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   15. associate-*l* N/A

       \[\leadsto c \cdot \left(\color{blue}{a \cdot \left(\frac{c}{{b}^{3}} \cdot \frac{-3}{8}\right)} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

5. Applied rewrites 95.7%

   \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.375, \frac{-0.5}{b}\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto c \cdot \left(a \cdot \left(\frac{c}{b \cdot \color{blue}{\left(b \cdot b\right)}} \cdot \frac{-3}{8}\right) + \frac{\frac{-1}{2}}{b}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto c \cdot \left(a \cdot \left(\frac{c}{\color{blue}{b \cdot \left(b \cdot b\right)}} \cdot \frac{-3}{8}\right) + \frac{\frac{-1}{2}}{b}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto c \cdot \left(a \cdot \left(\color{blue}{\frac{c}{b \cdot \left(b \cdot b\right)}} \cdot \frac{-3}{8}\right) + \frac{\frac{-1}{2}}{b}\right) \]

   4. lift-*.f64 N/A

      \[\leadsto c \cdot \left(a \cdot \color{blue}{\left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot \frac{-3}{8}\right)} + \frac{\frac{-1}{2}}{b}\right) \]

   5. lift-/.f64 N/A

      \[\leadsto c \cdot \left(a \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot \frac{-3}{8}\right) + \color{blue}{\frac{\frac{-1}{2}}{b}}\right) \]

   6. distribute-lft-in N/A

      \[\leadsto \color{blue}{c \cdot \left(a \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot \frac{-3}{8}\right)\right) + c \cdot \frac{\frac{-1}{2}}{b}} \]

   7. lift-/.f64 N/A

      \[\leadsto c \cdot \left(a \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot \frac{-3}{8}\right)\right) + c \cdot \color{blue}{\frac{\frac{-1}{2}}{b}} \]

   8. associate-*r/ N/A

      \[\leadsto c \cdot \left(a \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot \frac{-3}{8}\right)\right) + \color{blue}{\frac{c \cdot \frac{-1}{2}}{b}} \]

   9. associate-*l/ N/A

      \[\leadsto c \cdot \left(a \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot \frac{-3}{8}\right)\right) + \color{blue}{\frac{c}{b} \cdot \frac{-1}{2}} \]

   10. lift-/.f64 N/A

       \[\leadsto c \cdot \left(a \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot \frac{-3}{8}\right)\right) + \color{blue}{\frac{c}{b}} \cdot \frac{-1}{2} \]

   11. *-commutative N/A

       \[\leadsto c \cdot \left(a \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot \frac{-3}{8}\right)\right) + \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]

   12. lift-*.f64 N/A

       \[\leadsto c \cdot \left(a \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot \frac{-3}{8}\right)\right) + \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]

7. Applied rewrites 96.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}, a, \frac{c}{b \cdot -2}\right)} \]
8. Add Preprocessing

Alternative 3: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{a \cdot c}{a \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* a c) (* a (- (- b) (sqrt (fma a (* -3.0 c) (* b b)))))))

C

double code(double a, double b, double c) {
	return (a * c) / (a * (-b - sqrt(fma(a, (-3.0 * c), (b * b)))));
}

Julia

function code(a, b, c)
	return Float64(Float64(a * c) / Float64(a * Float64(Float64(-b) - sqrt(fma(a, Float64(-3.0 * c), Float64(b * b))))))
end

Wolfram

code[a_, b_, c_] := N[(N[(a * c), $MachinePrecision] / N[(a * N[((-b) - N[Sqrt[N[(a * N[(-3.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in c around inf

   \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{c \cdot \left(\frac{{b}^{2}}{c} - 3 \cdot a\right)}}}{3 \cdot a} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{c \cdot \left(\frac{{b}^{2}}{c} - 3 \cdot a\right)}}}{3 \cdot a} \]

   2. cancel-sign-sub-inv N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{c \cdot \color{blue}{\left(\frac{{b}^{2}}{c} + \left(\mathsf{neg}\left(3\right)\right) \cdot a\right)}}}{3 \cdot a} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{c \cdot \left(\frac{{b}^{2}}{c} + \color{blue}{-3} \cdot a\right)}}{3 \cdot a} \]

   4. +-commutative N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{c \cdot \color{blue}{\left(-3 \cdot a + \frac{{b}^{2}}{c}\right)}}}{3 \cdot a} \]

   5. *-commutative N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{c \cdot \left(\color{blue}{a \cdot -3} + \frac{{b}^{2}}{c}\right)}}{3 \cdot a} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{c \cdot \color{blue}{\mathsf{fma}\left(a, -3, \frac{{b}^{2}}{c}\right)}}}{3 \cdot a} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{c \cdot \mathsf{fma}\left(a, -3, \color{blue}{\frac{{b}^{2}}{c}}\right)}}{3 \cdot a} \]

   8. unpow2 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{c \cdot \mathsf{fma}\left(a, -3, \frac{\color{blue}{b \cdot b}}{c}\right)}}{3 \cdot a} \]

   9. lower-*.f64 17.5

      \[\leadsto \frac{\left(-b\right) + \sqrt{c \cdot \mathsf{fma}\left(a, -3, \frac{\color{blue}{b \cdot b}}{c}\right)}}{3 \cdot a} \]

5. Applied rewrites 17.5%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{c \cdot \mathsf{fma}\left(a, -3, \frac{b \cdot b}{c}\right)}}}{3 \cdot a} \]

7. Applied rewrites 18.0%

   \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(-b\right)}}}{3 \cdot a} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\left(c \cdot \color{blue}{\left(a \cdot -3\right)} + b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\left(c \cdot \left(a \cdot -3\right) + \color{blue}{b \cdot b}\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\left(c \cdot \left(a \cdot -3\right) + b \cdot b\right) - \color{blue}{b \cdot b}}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   4. associate--l+ N/A

      \[\leadsto \frac{\frac{\color{blue}{c \cdot \left(a \cdot -3\right) + \left(b \cdot b - b \cdot b\right)}}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\frac{c \cdot \color{blue}{\left(a \cdot -3\right)} + \left(b \cdot b - b \cdot b\right)}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   6. *-commutative N/A

      \[\leadsto \frac{\frac{c \cdot \color{blue}{\left(-3 \cdot a\right)} + \left(b \cdot b - b \cdot b\right)}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{c \cdot \left(\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a\right) + \left(b \cdot b - b \cdot b\right)}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   8. distribute-lft-neg-in N/A

      \[\leadsto \frac{\frac{c \cdot \color{blue}{\left(\mathsf{neg}\left(3 \cdot a\right)\right)} + \left(b \cdot b - b \cdot b\right)}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{c \cdot \left(\mathsf{neg}\left(\color{blue}{3 \cdot a}\right)\right) + \left(b \cdot b - b \cdot b\right)}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   10. *-commutative N/A

       \[\leadsto \frac{\frac{\color{blue}{\left(\mathsf{neg}\left(3 \cdot a\right)\right) \cdot c} + \left(b \cdot b - b \cdot b\right)}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   11. +-inverses N/A

       \[\leadsto \frac{\frac{\left(\mathsf{neg}\left(3 \cdot a\right)\right) \cdot c + \color{blue}{0}}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   12. lower-fma.f64 N/A

       \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(3 \cdot a\right), c, 0\right)}}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{3 \cdot a}\right), c, 0\right)}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   14. distribute-lft-neg-in N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot a}, c, 0\right)}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   15. metadata-eval N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{-3} \cdot a, c, 0\right)}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   16. *-commutative N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{a \cdot -3}, c, 0\right)}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

   17. lift-*.f64 99.5

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{a \cdot -3}, c, 0\right)}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(-b\right)}}{3 \cdot a} \]

9. Applied rewrites 99.5%

   \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(a \cdot -3, c, 0\right)}}{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - \left(-b\right)}}{3 \cdot a} \]

11. Applied rewrites 99.4%

    \[\leadsto \color{blue}{\frac{a \cdot c}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \cdot -1} \]

12. Final simplification 99.4%

    \[\leadsto \frac{a \cdot c}{a \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \]
13. Add Preprocessing

Alternative 4: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (fma a (/ (* (* c c) -0.375) (* b b)) (* c -0.5)) b))

C

double code(double a, double b, double c) {
	return fma(a, (((c * c) * -0.375) / (b * b)), (c * -0.5)) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(a, Float64(Float64(Float64(c * c) * -0.375) / Float64(b * b)), Float64(c * -0.5)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 17.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

5. Applied rewrites 96.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}} \]
6. Add Preprocessing

Alternative 5: 95.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \mathsf{fma}\left(-0.375, \frac{a \cdot c}{b \cdot b}, -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* c (fma -0.375 (/ (* a c) (* b b)) -0.5)) b))

C

double code(double a, double b, double c) {
	return (c * fma(-0.375, ((a * c) / (b * b)), -0.5)) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(c * fma(-0.375, Float64(Float64(a * c) / Float64(b * b)), -0.5)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(-0.375 * N[(N[(a * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 17.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

5. Applied rewrites 96.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}} \]

6. Taylor expanded in c around 0

   \[\leadsto \frac{\color{blue}{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}}{b} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}}{b} \]

   2. sub-neg N/A

      \[\leadsto \frac{c \cdot \color{blue}{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}}{b} \]

   3. metadata-eval N/A

      \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} + \color{blue}{\frac{-1}{2}}\right)}{b} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{c \cdot \color{blue}{\mathsf{fma}\left(\frac{-3}{8}, \frac{a \cdot c}{{b}^{2}}, \frac{-1}{2}\right)}}{b} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(\frac{-3}{8}, \color{blue}{\frac{a \cdot c}{{b}^{2}}}, \frac{-1}{2}\right)}{b} \]

   6. *-commutative N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(\frac{-3}{8}, \frac{\color{blue}{c \cdot a}}{{b}^{2}}, \frac{-1}{2}\right)}{b} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(\frac{-3}{8}, \frac{\color{blue}{c \cdot a}}{{b}^{2}}, \frac{-1}{2}\right)}{b} \]

   8. unpow2 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(\frac{-3}{8}, \frac{c \cdot a}{\color{blue}{b \cdot b}}, \frac{-1}{2}\right)}{b} \]

   9. lower-*.f64 96.0

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(-0.375, \frac{c \cdot a}{\color{blue}{b \cdot b}}, -0.5\right)}{b} \]

8. Applied rewrites 96.0%

   \[\leadsto \frac{\color{blue}{c \cdot \mathsf{fma}\left(-0.375, \frac{c \cdot a}{b \cdot b}, -0.5\right)}}{b} \]

9. Final simplification 96.0%

   \[\leadsto \frac{c \cdot \mathsf{fma}\left(-0.375, \frac{a \cdot c}{b \cdot b}, -0.5\right)}{b} \]
10. Add Preprocessing

Alternative 6: 94.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{\mathsf{fma}\left(-0.375, \frac{a \cdot c}{b \cdot b}, -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (/ (fma -0.375 (/ (* a c) (* b b)) -0.5) b)))

C

double code(double a, double b, double c) {
	return c * (fma(-0.375, ((a * c) / (b * b)), -0.5) / b);
}

Julia

function code(a, b, c)
	return Float64(c * Float64(fma(-0.375, Float64(Float64(a * c) / Float64(b * b)), -0.5) / b))
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(-0.375 * N[(N[(a * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto c \cdot \color{blue}{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}}\right) + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)} \]

   3. associate-*r/ N/A

      \[\leadsto c \cdot \color{blue}{\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{3}}} + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto c \cdot \frac{\color{blue}{\left(\frac{-3}{8} \cdot a\right) \cdot c}}{{b}^{3}} + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \]

   5. associate-*l/ N/A

      \[\leadsto c \cdot \color{blue}{\left(\frac{\frac{-3}{8} \cdot a}{{b}^{3}} \cdot c\right)} + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto c \cdot \left(\color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right)} \cdot c\right) + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \]

   7. distribute-lft-in N/A

      \[\leadsto \color{blue}{c \cdot \left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \color{blue}{c \cdot \left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

   9. associate-*r/ N/A

      \[\leadsto c \cdot \left(\color{blue}{\frac{\frac{-3}{8} \cdot a}{{b}^{3}}} \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   10. associate-*l/ N/A

       \[\leadsto c \cdot \left(\color{blue}{\frac{\left(\frac{-3}{8} \cdot a\right) \cdot c}{{b}^{3}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   11. associate-*r* N/A

       \[\leadsto c \cdot \left(\frac{\color{blue}{\frac{-3}{8} \cdot \left(a \cdot c\right)}}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   12. associate-*r/ N/A

       \[\leadsto c \cdot \left(\color{blue}{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto c \cdot \left(\color{blue}{\frac{a \cdot c}{{b}^{3}} \cdot \frac{-3}{8}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   14. associate-/l* N/A

       \[\leadsto c \cdot \left(\color{blue}{\left(a \cdot \frac{c}{{b}^{3}}\right)} \cdot \frac{-3}{8} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   15. associate-*l* N/A

       \[\leadsto c \cdot \left(\color{blue}{a \cdot \left(\frac{c}{{b}^{3}} \cdot \frac{-3}{8}\right)} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

5. Applied rewrites 95.7%

   \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.375, \frac{-0.5}{b}\right)} \]

6. Taylor expanded in b around inf

   \[\leadsto c \cdot \color{blue}{\frac{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}}{b}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto c \cdot \color{blue}{\frac{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}}{b}} \]

   2. sub-neg N/A

      \[\leadsto c \cdot \frac{\color{blue}{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{b} \]

   3. metadata-eval N/A

      \[\leadsto c \cdot \frac{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} + \color{blue}{\frac{-1}{2}}}{b} \]

   4. lower-fma.f64 N/A

      \[\leadsto c \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{-3}{8}, \frac{a \cdot c}{{b}^{2}}, \frac{-1}{2}\right)}}{b} \]

   5. lower-/.f64 N/A

      \[\leadsto c \cdot \frac{\mathsf{fma}\left(\frac{-3}{8}, \color{blue}{\frac{a \cdot c}{{b}^{2}}}, \frac{-1}{2}\right)}{b} \]

   6. lower-*.f64 N/A

      \[\leadsto c \cdot \frac{\mathsf{fma}\left(\frac{-3}{8}, \frac{\color{blue}{a \cdot c}}{{b}^{2}}, \frac{-1}{2}\right)}{b} \]

   7. unpow2 N/A

      \[\leadsto c \cdot \frac{\mathsf{fma}\left(\frac{-3}{8}, \frac{a \cdot c}{\color{blue}{b \cdot b}}, \frac{-1}{2}\right)}{b} \]

   8. lower-*.f64 95.6

      \[\leadsto c \cdot \frac{\mathsf{fma}\left(-0.375, \frac{a \cdot c}{\color{blue}{b \cdot b}}, -0.5\right)}{b} \]

8. Applied rewrites 95.6%

   \[\leadsto c \cdot \color{blue}{\frac{\mathsf{fma}\left(-0.375, \frac{a \cdot c}{b \cdot b}, -0.5\right)}{b}} \]
9. Add Preprocessing

Alternative 7: 90.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* -0.5 (/ c b)))

C

double code(double a, double b, double c) {
	return -0.5 * (c / b);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-0.5d0) * (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return -0.5 * (c / b);
}

Python

def code(a, b, c):
	return -0.5 * (c / b)

Julia

function code(a, b, c)
	return Float64(-0.5 * Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.5 * (c / b);
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]

   2. lower-/.f64 91.0

      \[\leadsto -0.5 \cdot \color{blue}{\frac{c}{b}} \]

5. Applied rewrites 91.0%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))